What Are the Frequency Components of an FM Wave?

• H_man
In summary, The individual frequency components of a wave with the equation y(t) = Acos(W0t+W1COS(W2t)t ) can be shown through a Fourier transform, however integrating the cosine term can be challenging. Alternatively, a DFT using Matlab can be used. Expanding the equation in terms of trigonometric identities does not lead to a solution, but the use of Bessel Functions can provide a solution.
H_man

Homework Statement

I wanted to show that a particular wave is composed of the 4 frequencies

W = W0 +/- W1 +/- W2

The equation for this FM wave is y(t) = Acos(W0t+W1COS(W2t)t )

I tried showing the different frequency components by performing a Fourier transform on the equation y(t). However I got a bit stuck with integrating the cosine term as part of the exponential function. I thought about expanding it as a Taylor series but that doesn't really help.

If this was a practical problem I would just use Matlab and a DFT and hey presto. But trying to do this analytically is posing a problem.

Expanding in terms of trigonometric identities to show the point also doesn't seem to get me closer to a solution.

y(t) = Acos(W0t+W1COS(W2t)t ) =

= A{cos(W0t)cos(W1COS(W2t)t ) -
sin(W0t)sin(W1sin(W2t)t )}

we are given that W1t<<1
and W12t -> 0
so if we assume cos(W1COS(W2t)t )~1

Then we have:

= A{cos(W0t)-sin(W0t)W1cos(W2t)t )}

= A{cos(W0t) - sin(W0t+W2t)W1 +
W1 cos(W0t)sin(W2t)

But I don't see a solution from here...

Its okay. I just found the solution. It involves Bessel Functions...

1. What is the frequency range of FM waves?

The frequency range of FM waves typically falls between 88 MHz to 108 MHz. This range is referred to as the "very high frequency" (VHF) band.

2. How are the frequency components of FM waves different from AM waves?

The frequency components of FM waves are different from AM waves in that they vary in a continuous manner. In FM waves, the amplitude remains constant while the frequency changes, whereas in AM waves, the frequency remains constant while the amplitude changes.

3. How do frequency components affect the sound quality of FM waves?

The frequency components of FM waves play a crucial role in determining the sound quality. A wider range of frequency components allows for a greater range of sounds to be transmitted, resulting in a higher fidelity and clearer sound quality.

4. Can FM waves have negative frequency components?

No, FM waves cannot have negative frequency components. The frequency of an FM wave is always positive and increases in proportion to the amplitude of the modulating signal.

5. What are the limitations of frequency components in FM waves?

One limitation of frequency components in FM waves is that there is a limited range of frequencies that can be transmitted within the allocated frequency band. Additionally, interference from other signals and atmospheric conditions can affect the transmission of frequency components and impact the overall sound quality.

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